LMFDB - Embedded newform 1200.4.a.v.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,4,Mod(1,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1200.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$1200 = 2^{4} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,0,0,-23,0,9,0,30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$70.8022920069$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1200.1

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

f(q)

=

q+3.00000 q^{3} -23.0000 q^{7} +9.00000 q^{9} +30.0000 q^{11} +29.0000 q^{13} +78.0000 q^{17} -149.000 q^{19} -69.0000 q^{21} -150.000 q^{23} +27.0000 q^{27} -234.000 q^{29} +217.000 q^{31} +90.0000 q^{33} +146.000 q^{37} +87.0000 q^{39} -156.000 q^{41} +433.000 q^{43} -30.0000 q^{47} +186.000 q^{49} +234.000 q^{51} -552.000 q^{53} -447.000 q^{57} +270.000 q^{59} +275.000 q^{61} -207.000 q^{63} -803.000 q^{67} -450.000 q^{69} -660.000 q^{71} -646.000 q^{73} -690.000 q^{77} -992.000 q^{79} +81.0000 q^{81} +846.000 q^{83} -702.000 q^{87} -1488.00 q^{89} -667.000 q^{91} +651.000 q^{93} -319.000 q^{97} +270.000 q^{99} +O(q^{100})

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−23.0000	−1.24188	−0.620942	−	0.783857i	$-0.713250\pi$
$7$	−23.0000	−1.24188	−0.620942	+	0.783857i	$0.713250\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	30.0000	0.822304	0.411152	−	0.911567i	$-0.365127\pi$
$11$	30.0000	0.822304	0.411152	+	0.911567i	$0.365127\pi$
$12$	0	0
$13$	29.0000	0.618704	0.309352	−	0.950948i	$-0.399888\pi$
$13$	29.0000	0.618704	0.309352	+	0.950948i	$0.399888\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	78.0000	1.11281	0.556405	−	0.830911i	$-0.312180\pi$
$17$	78.0000	1.11281	0.556405	+	0.830911i	$0.312180\pi$
$18$	0	0
$19$	−149.000	−1.79910	−0.899551	−	0.436815i	$-0.856106\pi$
$19$	−149.000	−1.79910	−0.899551	+	0.436815i	$0.856106\pi$
$20$	0	0
$21$	−69.0000	−0.717002
$22$	0	0
$23$	−150.000	−1.35988	−0.679938	−	0.733269i	$-0.737993\pi$
$23$	−150.000	−1.35988	−0.679938	+	0.733269i	$0.737993\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−234.000	−1.49837	−0.749185	−	0.662361i	$-0.769554\pi$
$29$	−234.000	−1.49837	−0.749185	+	0.662361i	$0.769554\pi$
$30$	0	0
$31$	217.000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	217.000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	90.0000	0.474757
$34$	0	0
$35$	0	0
$36$	0	0
$37$	146.000	0.648710	0.324355	−	0.945936i	$-0.394853\pi$
$37$	146.000	0.648710	0.324355	+	0.945936i	$0.394853\pi$
$38$	0	0
$39$	87.0000	0.357209
$40$	0	0
$41$	−156.000	−0.594222	−0.297111	−	0.954843i	$-0.596023\pi$
$41$	−156.000	−0.594222	−0.297111	+	0.954843i	$0.596023\pi$
$42$	0	0
$43$	433.000	1.53563	0.767813	−	0.640675i	$-0.221345\pi$
$43$	433.000	1.53563	0.767813	+	0.640675i	$0.221345\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−30.0000	−0.0931053	−0.0465527	−	0.998916i	$-0.514824\pi$
$47$	−30.0000	−0.0931053	−0.0465527	+	0.998916i	$0.514824\pi$
$48$	0	0
$49$	186.000	0.542274
$50$	0	0
$51$	234.000	0.642481
$52$	0	0
$53$	−552.000	−1.43062	−0.715312	−	0.698806i	$-0.753715\pi$
$53$	−552.000	−1.43062	−0.715312	+	0.698806i	$0.753715\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−447.000	−1.03871
$58$	0	0
$59$	270.000	0.595780	0.297890	−	0.954600i	$-0.403717\pi$
$59$	270.000	0.595780	0.297890	+	0.954600i	$0.403717\pi$
$60$	0	0
$61$	275.000	0.577215	0.288608	−	0.957447i	$-0.406808\pi$
$61$	275.000	0.577215	0.288608	+	0.957447i	$0.406808\pi$
$62$	0	0
$63$	−207.000	−0.413961
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−803.000	−1.46421	−0.732105	−	0.681192i	$-0.761462\pi$
$67$	−803.000	−1.46421	−0.732105	+	0.681192i	$0.761462\pi$
$68$	0	0
$69$	−450.000	−0.785125
$70$	0	0
$71$	−660.000	−1.10321	−0.551603	−	0.834107i	$-0.685984\pi$
$71$	−660.000	−1.10321	−0.551603	+	0.834107i	$0.685984\pi$
$72$	0	0
$73$	−646.000	−1.03573	−0.517867	−	0.855461i	$-0.673274\pi$
$73$	−646.000	−1.03573	−0.517867	+	0.855461i	$0.673274\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−690.000	−1.02121
$78$	0	0
$79$	−992.000	−1.41277	−0.706384	−	0.707829i	$-0.749675\pi$
$79$	−992.000	−1.41277	−0.706384	+	0.707829i	$0.749675\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	846.000	1.11880	0.559401	−	0.828897i	$-0.311031\pi$
$83$	846.000	1.11880	0.559401	+	0.828897i	$0.311031\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−702.000	−0.865084
$88$	0	0
$89$	−1488.00	−1.77222	−0.886111	−	0.463474i	$-0.846603\pi$
$89$	−1488.00	−1.77222	−0.886111	+	0.463474i	$0.846603\pi$
$90$	0	0
$91$	−667.000	−0.768358
$92$	0	0
$93$	651.000	0.725866
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−319.000	−0.333913	−0.166956	−	0.985964i	$-0.553394\pi$
$97$	−319.000	−0.333913	−0.166956	+	0.985964i	$0.553394\pi$
$98$	0	0
$99$	270.000	0.274101
$100$	0	0
$101$	−792.000	−0.780267	−0.390133	−	0.920758i	$-0.627571\pi$
$101$	−792.000	−0.780267	−0.390133	+	0.920758i	$0.627571\pi$
$102$	0	0
$103$	−812.000	−0.776784	−0.388392	−	0.921494i	$-0.626969\pi$
$103$	−812.000	−0.776784	−0.388392	+	0.921494i	$0.626969\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1416.00	1.27934	0.639672	−	0.768648i	$-0.279070\pi$
$107$	1416.00	1.27934	0.639672	+	0.768648i	$0.279070\pi$
$108$	0	0
$109$	−55.0000	−0.0483307	−0.0241653	−	0.999708i	$-0.507693\pi$
$109$	−55.0000	−0.0483307	−0.0241653	+	0.999708i	$0.507693\pi$
$110$	0	0
$111$	438.000	0.374533
$112$	0	0
$113$	1404.00	1.16882	0.584412	−	0.811457i	$-0.301325\pi$
$113$	1404.00	1.16882	0.584412	+	0.811457i	$0.301325\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	261.000	0.206235
$118$	0	0
$119$	−1794.00	−1.38198
$120$	0	0
$121$	−431.000	−0.323817
$122$	0	0
$123$	−468.000	−0.343074
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1280.00	−0.894344	−0.447172	−	0.894448i	$-0.647569\pi$
$127$	−1280.00	−0.894344	−0.447172	+	0.894448i	$0.647569\pi$
$128$	0	0
$129$	1299.00	0.886594
$130$	0	0
$131$	−480.000	−0.320136	−0.160068	−	0.987106i	$-0.551171\pi$
$131$	−480.000	−0.320136	−0.160068	+	0.987106i	$0.551171\pi$
$132$	0	0
$133$	3427.00	2.23428
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−282.000	−0.175860	−0.0879302	−	0.996127i	$-0.528025\pi$
$137$	−282.000	−0.175860	−0.0879302	+	0.996127i	$0.528025\pi$
$138$	0	0
$139$	−1604.00	−0.978773	−0.489387	−	0.872067i	$-0.662779\pi$
$139$	−1604.00	−0.978773	−0.489387	+	0.872067i	$0.662779\pi$
$140$	0	0
$141$	−90.0000	−0.0537544
$142$	0	0
$143$	870.000	0.508763
$144$	0	0
$145$	0	0
$146$	0	0
$147$	558.000	0.313082
$148$	0	0
$149$	−774.000	−0.425561	−0.212780	−	0.977100i	$-0.568252\pi$
$149$	−774.000	−0.425561	−0.212780	+	0.977100i	$0.568252\pi$
$150$	0	0
$151$	−293.000	−0.157907	−0.0789536	−	0.996878i	$-0.525158\pi$
$151$	−293.000	−0.157907	−0.0789536	+	0.996878i	$0.525158\pi$
$152$	0	0
$153$	702.000	0.370937
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1729.00	−0.878912	−0.439456	−	0.898264i	$-0.644829\pi$
$157$	−1729.00	−0.878912	−0.439456	+	0.898264i	$0.644829\pi$
$158$	0	0
$159$	−1656.00	−0.825971
$160$	0	0
$161$	3450.00	1.68881
$162$	0	0
$163$	1123.00	0.539633	0.269816	−	0.962912i	$-0.413037\pi$
$163$	1123.00	0.539633	0.269816	+	0.962912i	$0.413037\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1200.00	−0.556041	−0.278020	−	0.960575i	$-0.589678\pi$
$167$	−1200.00	−0.556041	−0.278020	+	0.960575i	$0.589678\pi$
$168$	0	0
$169$	−1356.00	−0.617205
$170$	0	0
$171$	−1341.00	−0.599701
$172$	0	0
$173$	−1734.00	−0.762044	−0.381022	−	0.924566i	$-0.624428\pi$
$173$	−1734.00	−0.762044	−0.381022	+	0.924566i	$0.624428\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	810.000	0.343974
$178$	0	0
$179$	−2586.00	−1.07981	−0.539907	−	0.841725i	$-0.681541\pi$
$179$	−2586.00	−1.07981	−0.539907	+	0.841725i	$0.681541\pi$
$180$	0	0
$181$	−3931.00	−1.61430	−0.807152	−	0.590344i	$-0.798992\pi$
$181$	−3931.00	−1.61430	−0.807152	+	0.590344i	$0.798992\pi$
$182$	0	0
$183$	825.000	0.333255
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2340.00	0.915068
$188$	0	0
$189$	−621.000	−0.239001
$190$	0	0
$191$	−1566.00	−0.593255	−0.296628	−	0.954993i	$-0.595862\pi$
$191$	−1566.00	−0.593255	−0.296628	+	0.954993i	$0.595862\pi$
$192$	0	0
$193$	2291.00	0.854455	0.427227	−	0.904144i	$-0.359490\pi$
$193$	2291.00	0.854455	0.427227	+	0.904144i	$0.359490\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2142.00	0.774676	0.387338	−	0.921938i	$-0.373395\pi$
$197$	2142.00	0.774676	0.387338	+	0.921938i	$0.373395\pi$
$198$	0	0
$199$	4903.00	1.74656	0.873278	−	0.487223i	$-0.161990\pi$
$199$	4903.00	1.74656	0.873278	+	0.487223i	$0.161990\pi$
$200$	0	0
$201$	−2409.00	−0.845362
$202$	0	0
$203$	5382.00	1.86080
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1350.00	−0.453292
$208$	0	0
$209$	−4470.00	−1.47941
$210$	0	0
$211$	−605.000	−0.197393	−0.0986965	−	0.995118i	$-0.531467\pi$
$211$	−605.000	−0.197393	−0.0986965	+	0.995118i	$0.531467\pi$
$212$	0	0
$213$	−1980.00	−0.636936
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4991.00	−1.56134
$218$	0	0
$219$	−1938.00	−0.597981
$220$	0	0
$221$	2262.00	0.688500
$222$	0	0
$223$	145.000	0.0435422	0.0217711	−	0.999763i	$-0.493069\pi$
$223$	145.000	0.0435422	0.0217711	+	0.999763i	$0.493069\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2964.00	−0.866641	−0.433321	−	0.901240i	$-0.642658\pi$
$227$	−2964.00	−0.866641	−0.433321	+	0.901240i	$0.642658\pi$
$228$	0	0
$229$	−5635.00	−1.62608	−0.813038	−	0.582211i	$-0.802188\pi$
$229$	−5635.00	−1.62608	−0.813038	+	0.582211i	$0.802188\pi$
$230$	0	0
$231$	−2070.00	−0.589593
$232$	0	0
$233$	4164.00	1.17078	0.585392	−	0.810750i	$-0.300941\pi$
$233$	4164.00	1.17078	0.585392	+	0.810750i	$0.300941\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2976.00	−0.815662
$238$	0	0
$239$	1944.00	0.526138	0.263069	−	0.964777i	$-0.415265\pi$
$239$	1944.00	0.526138	0.263069	+	0.964777i	$0.415265\pi$
$240$	0	0
$241$	857.000	0.229063	0.114532	−	0.993420i	$-0.463463\pi$
$241$	857.000	0.229063	0.114532	+	0.993420i	$0.463463\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4321.00	−1.11311
$248$	0	0
$249$	2538.00	0.645941
$250$	0	0
$251$	3924.00	0.986776	0.493388	−	0.869809i	$-0.335758\pi$
$251$	3924.00	0.986776	0.493388	+	0.869809i	$0.335758\pi$
$252$	0	0
$253$	−4500.00	−1.11823
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2844.00	−0.690287	−0.345144	−	0.938550i	$-0.612170\pi$
$257$	−2844.00	−0.690287	−0.345144	+	0.938550i	$0.612170\pi$
$258$	0	0
$259$	−3358.00	−0.805621
$260$	0	0
$261$	−2106.00	−0.499456
$262$	0	0
$263$	6060.00	1.42082	0.710410	−	0.703788i	$-0.248510\pi$
$263$	6060.00	1.42082	0.710410	+	0.703788i	$0.248510\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4464.00	−1.02319
$268$	0	0
$269$	3906.00	0.885327	0.442664	−	0.896688i	$-0.354034\pi$
$269$	3906.00	0.885327	0.442664	+	0.896688i	$0.354034\pi$
$270$	0	0
$271$	−2144.00	−0.480586	−0.240293	−	0.970700i	$-0.577243\pi$
$271$	−2144.00	−0.480586	−0.240293	+	0.970700i	$0.577243\pi$
$272$	0	0
$273$	−2001.00	−0.443612
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2321.00	0.503449	0.251725	−	0.967799i	$-0.419002\pi$
$277$	2321.00	0.503449	0.251725	+	0.967799i	$0.419002\pi$
$278$	0	0
$279$	1953.00	0.419079
$280$	0	0
$281$	−6822.00	−1.44828	−0.724140	−	0.689654i	$-0.757763\pi$
$281$	−6822.00	−1.44828	−0.724140	+	0.689654i	$0.757763\pi$
$282$	0	0
$283$	−4049.00	−0.850488	−0.425244	−	0.905079i	$-0.639812\pi$
$283$	−4049.00	−0.850488	−0.425244	+	0.905079i	$0.639812\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3588.00	0.737955
$288$	0	0
$289$	1171.00	0.238347
$290$	0	0
$291$	−957.000	−0.192785
$292$	0	0
$293$	2238.00	0.446230	0.223115	−	0.974792i	$-0.428377\pi$
$293$	2238.00	0.446230	0.223115	+	0.974792i	$0.428377\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	810.000	0.158252
$298$	0	0
$299$	−4350.00	−0.841361
$300$	0	0
$301$	−9959.00	−1.90707
$302$	0	0
$303$	−2376.00	−0.450487
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1385.00	−0.257479	−0.128740	−	0.991678i	$-0.541093\pi$
$307$	−1385.00	−0.257479	−0.128740	+	0.991678i	$0.541093\pi$
$308$	0	0
$309$	−2436.00	−0.448476
$310$	0	0
$311$	5670.00	1.03381	0.516907	−	0.856042i	$-0.327083\pi$
$311$	5670.00	1.03381	0.516907	+	0.856042i	$0.327083\pi$
$312$	0	0
$313$	−421.000	−0.0760266	−0.0380133	−	0.999277i	$-0.512103\pi$
$313$	−421.000	−0.0760266	−0.0380133	+	0.999277i	$0.512103\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9984.00	1.76895	0.884475	−	0.466587i	$-0.154517\pi$
$317$	9984.00	1.76895	0.884475	+	0.466587i	$0.154517\pi$
$318$	0	0
$319$	−7020.00	−1.23211
$320$	0	0
$321$	4248.00	0.738630
$322$	0	0
$323$	−11622.0	−2.00206
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−165.000	−0.0279037
$328$	0	0
$329$	690.000	0.115626
$330$	0	0
$331$	4228.00	0.702090	0.351045	−	0.936359i	$-0.385826\pi$
$331$	4228.00	0.702090	0.351045	+	0.936359i	$0.385826\pi$
$332$	0	0
$333$	1314.00	0.216237
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5393.00	0.871737	0.435869	−	0.900010i	$-0.356441\pi$
$337$	5393.00	0.871737	0.435869	+	0.900010i	$0.356441\pi$
$338$	0	0
$339$	4212.00	0.674821
$340$	0	0
$341$	6510.00	1.03383
$342$	0	0
$343$	3611.00	0.568442
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7914.00	1.22434	0.612170	−	0.790726i	$-0.290297\pi$
$347$	7914.00	1.22434	0.612170	+	0.790726i	$0.290297\pi$
$348$	0	0
$349$	1010.00	0.154911	0.0774557	−	0.996996i	$-0.475320\pi$
$349$	1010.00	0.154911	0.0774557	+	0.996996i	$0.475320\pi$
$350$	0	0
$351$	783.000	0.119070
$352$	0	0
$353$	4722.00	0.711974	0.355987	−	0.934491i	$-0.384145\pi$
$353$	4722.00	0.711974	0.355987	+	0.934491i	$0.384145\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5382.00	−0.797887
$358$	0	0
$359$	−6204.00	−0.912074	−0.456037	−	0.889961i	$-0.650732\pi$
$359$	−6204.00	−0.912074	−0.456037	+	0.889961i	$0.650732\pi$
$360$	0	0
$361$	15342.0	2.23677
$362$	0	0
$363$	−1293.00	−0.186956
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1361.00	−0.193579	−0.0967897	−	0.995305i	$-0.530857\pi$
$367$	−1361.00	−0.193579	−0.0967897	+	0.995305i	$0.530857\pi$
$368$	0	0
$369$	−1404.00	−0.198074
$370$	0	0
$371$	12696.0	1.77667
$372$	0	0
$373$	−913.000	−0.126738	−0.0633691	−	0.997990i	$-0.520185\pi$
$373$	−913.000	−0.126738	−0.0633691	+	0.997990i	$0.520185\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6786.00	−0.927047
$378$	0	0
$379$	8881.00	1.20366	0.601829	−	0.798625i	$-0.294439\pi$
$379$	8881.00	1.20366	0.601829	+	0.798625i	$0.294439\pi$
$380$	0	0
$381$	−3840.00	−0.516350
$382$	0	0
$383$	−5460.00	−0.728441	−0.364221	−	0.931313i	$-0.618665\pi$
$383$	−5460.00	−0.728441	−0.364221	+	0.931313i	$0.618665\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3897.00	0.511875
$388$	0	0
$389$	−13884.0	−1.80963	−0.904816	−	0.425803i	$-0.859992\pi$
$389$	−13884.0	−1.80963	−0.904816	+	0.425803i	$0.859992\pi$
$390$	0	0
$391$	−11700.0	−1.51328
$392$	0	0
$393$	−1440.00	−0.184831
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3781.00	−0.477992	−0.238996	−	0.971021i	$-0.576818\pi$
$397$	−3781.00	−0.477992	−0.238996	+	0.971021i	$0.576818\pi$
$398$	0	0
$399$	10281.0	1.28996
$400$	0	0
$401$	9024.00	1.12378	0.561892	−	0.827211i	$-0.310074\pi$
$401$	9024.00	1.12378	0.561892	+	0.827211i	$0.310074\pi$
$402$	0	0
$403$	6293.00	0.777858
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4380.00	0.533436
$408$	0	0
$409$	14789.0	1.78794	0.893972	−	0.448123i	$-0.147907\pi$
$409$	14789.0	1.78794	0.893972	+	0.448123i	$0.147907\pi$
$410$	0	0
$411$	−846.000	−0.101533
$412$	0	0
$413$	−6210.00	−0.739889
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4812.00	−0.565095
$418$	0	0
$419$	−9840.00	−1.14729	−0.573646	−	0.819103i	$-0.694472\pi$
$419$	−9840.00	−1.14729	−0.573646	+	0.819103i	$0.694472\pi$
$420$	0	0
$421$	5510.00	0.637865	0.318932	−	0.947778i	$-0.396676\pi$
$421$	5510.00	0.637865	0.318932	+	0.947778i	$0.396676\pi$
$422$	0	0
$423$	−270.000	−0.0310351
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6325.00	−0.716834
$428$	0	0
$429$	2610.00	0.293734
$430$	0	0
$431$	−11070.0	−1.23718	−0.618588	−	0.785715i	$-0.712295\pi$
$431$	−11070.0	−1.23718	−0.618588	+	0.785715i	$0.712295\pi$
$432$	0	0
$433$	−12133.0	−1.34659	−0.673297	−	0.739373i	$-0.735122\pi$
$433$	−12133.0	−1.34659	−0.673297	+	0.739373i	$0.735122\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22350.0	2.44656
$438$	0	0
$439$	1873.00	0.203630	0.101815	−	0.994803i	$-0.467535\pi$
$439$	1873.00	0.203630	0.101815	+	0.994803i	$0.467535\pi$
$440$	0	0
$441$	1674.00	0.180758
$442$	0	0
$443$	576.000	0.0617756	0.0308878	−	0.999523i	$-0.490167\pi$
$443$	576.000	0.0617756	0.0308878	+	0.999523i	$0.490167\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2322.00	−0.245698
$448$	0	0
$449$	−4884.00	−0.513341	−0.256671	−	0.966499i	$-0.582626\pi$
$449$	−4884.00	−0.513341	−0.256671	+	0.966499i	$0.582626\pi$
$450$	0	0
$451$	−4680.00	−0.488631
$452$	0	0
$453$	−879.000	−0.0911678
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15802.0	−1.61748	−0.808738	−	0.588169i	$-0.799849\pi$
$457$	−15802.0	−1.61748	−0.808738	+	0.588169i	$0.799849\pi$
$458$	0	0
$459$	2106.00	0.214160
$460$	0	0
$461$	−15360.0	−1.55181	−0.775907	−	0.630847i	$-0.782708\pi$
$461$	−15360.0	−1.55181	−0.775907	+	0.630847i	$0.782708\pi$
$462$	0	0
$463$	−1712.00	−0.171843	−0.0859216	−	0.996302i	$-0.527383\pi$
$463$	−1712.00	−0.171843	−0.0859216	+	0.996302i	$0.527383\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16278.0	1.61297	0.806484	−	0.591256i	$-0.201368\pi$
$467$	16278.0	1.61297	0.806484	+	0.591256i	$0.201368\pi$
$468$	0	0
$469$	18469.0	1.81838
$470$	0	0
$471$	−5187.00	−0.507440
$472$	0	0
$473$	12990.0	1.26275
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4968.00	−0.476874
$478$	0	0
$479$	14766.0	1.40851	0.704254	−	0.709948i	$-0.251281\pi$
$479$	14766.0	1.40851	0.704254	+	0.709948i	$0.251281\pi$
$480$	0	0
$481$	4234.00	0.401359
$482$	0	0
$483$	10350.0	0.975034
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3319.00	0.308826	0.154413	−	0.988006i	$-0.450651\pi$
$487$	3319.00	0.308826	0.154413	+	0.988006i	$0.450651\pi$
$488$	0	0
$489$	3369.00	0.311557
$490$	0	0
$491$	−11064.0	−1.01693	−0.508464	−	0.861083i	$-0.669786\pi$
$491$	−11064.0	−1.01693	−0.508464	+	0.861083i	$0.669786\pi$
$492$	0	0
$493$	−18252.0	−1.66740
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15180.0	1.37005
$498$	0	0
$499$	14131.0	1.26772	0.633858	−	0.773449i	$-0.281470\pi$
$499$	14131.0	1.26772	0.633858	+	0.773449i	$0.281470\pi$
$500$	0	0
$501$	−3600.00	−0.321030
$502$	0	0
$503$	−11988.0	−1.06266	−0.531331	−	0.847165i	$-0.678308\pi$
$503$	−11988.0	−1.06266	−0.531331	+	0.847165i	$0.678308\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4068.00	−0.356344
$508$	0	0
$509$	10806.0	0.940997	0.470499	−	0.882401i	$-0.344074\pi$
$509$	10806.0	0.940997	0.470499	+	0.882401i	$0.344074\pi$
$510$	0	0
$511$	14858.0	1.28626
$512$	0	0
$513$	−4023.00	−0.346237
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−900.000	−0.0765608
$518$	0	0
$519$	−5202.00	−0.439966
$520$	0	0
$521$	22578.0	1.89858	0.949290	−	0.314402i	$-0.101804\pi$
$521$	22578.0	1.89858	0.949290	+	0.314402i	$0.101804\pi$
$522$	0	0
$523$	−12065.0	−1.00873	−0.504365	−	0.863491i	$-0.668273\pi$
$523$	−12065.0	−1.00873	−0.504365	+	0.863491i	$0.668273\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16926.0	1.39907
$528$	0	0
$529$	10333.0	0.849264
$530$	0	0
$531$	2430.00	0.198593
$532$	0	0
$533$	−4524.00	−0.367648
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7758.00	−0.623431
$538$	0	0
$539$	5580.00	0.445914
$540$	0	0
$541$	−12055.0	−0.958013	−0.479006	−	0.877811i	$-0.659003\pi$
$541$	−12055.0	−0.958013	−0.479006	+	0.877811i	$0.659003\pi$
$542$	0	0
$543$	−11793.0	−0.932019
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6176.00	−0.482754	−0.241377	−	0.970431i	$-0.577599\pi$
$547$	−6176.00	−0.482754	−0.241377	+	0.970431i	$0.577599\pi$
$548$	0	0
$549$	2475.00	0.192405
$550$	0	0
$551$	34866.0	2.69572
$552$	0	0
$553$	22816.0	1.75449
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8274.00	0.629409	0.314704	−	0.949190i	$-0.398095\pi$
$557$	8274.00	0.629409	0.314704	+	0.949190i	$0.398095\pi$
$558$	0	0
$559$	12557.0	0.950098
$560$	0	0
$561$	7020.00	0.528315
$562$	0	0
$563$	−966.000	−0.0723127	−0.0361563	−	0.999346i	$-0.511511\pi$
$563$	−966.000	−0.0723127	−0.0361563	+	0.999346i	$0.511511\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1863.00	−0.137987
$568$	0	0
$569$	19002.0	1.40001	0.700005	−	0.714138i	$-0.253181\pi$
$569$	19002.0	1.40001	0.700005	+	0.714138i	$0.253181\pi$
$570$	0	0
$571$	−8645.00	−0.633594	−0.316797	−	0.948493i	$-0.602607\pi$
$571$	−8645.00	−0.633594	−0.316797	+	0.948493i	$0.602607\pi$
$572$	0	0
$573$	−4698.00	−0.342516
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10931.0	0.788672	0.394336	−	0.918966i	$-0.370975\pi$
$577$	10931.0	0.788672	0.394336	+	0.918966i	$0.370975\pi$
$578$	0	0
$579$	6873.00	0.493320
$580$	0	0
$581$	−19458.0	−1.38942
$582$	0	0
$583$	−16560.0	−1.17641
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8904.00	−0.626077	−0.313039	−	0.949740i	$-0.601347\pi$
$587$	−8904.00	−0.626077	−0.313039	+	0.949740i	$0.601347\pi$
$588$	0	0
$589$	−32333.0	−2.26190
$590$	0	0
$591$	6426.00	0.447259
$592$	0	0
$593$	8820.00	0.610782	0.305391	−	0.952227i	$-0.401213\pi$
$593$	8820.00	0.610782	0.305391	+	0.952227i	$0.401213\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14709.0	1.00837
$598$	0	0
$599$	−9804.00	−0.668749	−0.334374	−	0.942440i	$-0.608525\pi$
$599$	−9804.00	−0.668749	−0.334374	+	0.942440i	$0.608525\pi$
$600$	0	0
$601$	−23437.0	−1.59071	−0.795354	−	0.606146i	$-0.792715\pi$
$601$	−23437.0	−1.59071	−0.795354	+	0.606146i	$0.792715\pi$
$602$	0	0
$603$	−7227.00	−0.488070
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2648.00	−0.177066	−0.0885330	−	0.996073i	$-0.528218\pi$
$607$	−2648.00	−0.177066	−0.0885330	+	0.996073i	$0.528218\pi$
$608$	0	0
$609$	16146.0	1.07433
$610$	0	0
$611$	−870.000	−0.0576046
$612$	0	0
$613$	794.000	0.0523154	0.0261577	−	0.999658i	$-0.491673\pi$
$613$	794.000	0.0523154	0.0261577	+	0.999658i	$0.491673\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18720.0	1.22146	0.610728	−	0.791840i	$-0.290877\pi$
$617$	18720.0	1.22146	0.610728	+	0.791840i	$0.290877\pi$
$618$	0	0
$619$	8959.00	0.581733	0.290866	−	0.956764i	$-0.406056\pi$
$619$	8959.00	0.581733	0.290866	+	0.956764i	$0.406056\pi$
$620$	0	0
$621$	−4050.00	−0.261708
$622$	0	0
$623$	34224.0	2.20089
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−13410.0	−0.854137
$628$	0	0
$629$	11388.0	0.721891
$630$	0	0
$631$	12373.0	0.780604	0.390302	−	0.920687i	$-0.372371\pi$
$631$	12373.0	0.780604	0.390302	+	0.920687i	$0.372371\pi$
$632$	0	0
$633$	−1815.00	−0.113965
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5394.00	0.335507
$638$	0	0
$639$	−5940.00	−0.367735
$640$	0	0
$641$	24900.0	1.53431	0.767154	−	0.641463i	$-0.221672\pi$
$641$	24900.0	1.53431	0.767154	+	0.641463i	$0.221672\pi$
$642$	0	0
$643$	14668.0	0.899610	0.449805	−	0.893127i	$-0.351493\pi$
$643$	14668.0	0.899610	0.449805	+	0.893127i	$0.351493\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10788.0	0.655518	0.327759	−	0.944761i	$-0.393707\pi$
$647$	10788.0	0.655518	0.327759	+	0.944761i	$0.393707\pi$
$648$	0	0
$649$	8100.00	0.489912
$650$	0	0
$651$	−14973.0	−0.901441
$652$	0	0
$653$	−14214.0	−0.851817	−0.425909	−	0.904766i	$-0.640046\pi$
$653$	−14214.0	−0.851817	−0.425909	+	0.904766i	$0.640046\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5814.00	−0.345245
$658$	0	0
$659$	588.000	0.0347576	0.0173788	−	0.999849i	$-0.494468\pi$
$659$	588.000	0.0347576	0.0173788	+	0.999849i	$0.494468\pi$
$660$	0	0
$661$	−3166.00	−0.186298	−0.0931491	−	0.995652i	$-0.529693\pi$
$661$	−3166.00	−0.186298	−0.0931491	+	0.995652i	$0.529693\pi$
$662$	0	0
$663$	6786.00	0.397506
$664$	0	0
$665$	0	0
$666$	0	0
$667$	35100.0	2.03760
$668$	0	0
$669$	435.000	0.0251391
$670$	0	0
$671$	8250.00	0.474646
$672$	0	0
$673$	9182.00	0.525914	0.262957	−	0.964808i	$-0.415302\pi$
$673$	9182.00	0.525914	0.262957	+	0.964808i	$0.415302\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11742.0	0.666590	0.333295	−	0.942823i	$-0.391839\pi$
$677$	11742.0	0.666590	0.333295	+	0.942823i	$0.391839\pi$
$678$	0	0
$679$	7337.00	0.414681
$680$	0	0
$681$	−8892.00	−0.500356
$682$	0	0
$683$	6024.00	0.337485	0.168742	−	0.985660i	$-0.446029\pi$
$683$	6024.00	0.337485	0.168742	+	0.985660i	$0.446029\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16905.0	−0.938815
$688$	0	0
$689$	−16008.0	−0.885132
$690$	0	0
$691$	−9344.00	−0.514418	−0.257209	−	0.966356i	$-0.582803\pi$
$691$	−9344.00	−0.514418	−0.257209	+	0.966356i	$0.582803\pi$
$692$	0	0
$693$	−6210.00	−0.340402
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12168.0	−0.661257
$698$	0	0
$699$	12492.0	0.675953
$700$	0	0
$701$	21234.0	1.14408	0.572038	−	0.820227i	$-0.306153\pi$
$701$	21234.0	1.14408	0.572038	+	0.820227i	$0.306153\pi$
$702$	0	0
$703$	−21754.0	−1.16709
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18216.0	0.969000
$708$	0	0
$709$	−1723.00	−0.0912675	−0.0456337	−	0.998958i	$-0.514531\pi$
$709$	−1723.00	−0.0912675	−0.0456337	+	0.998958i	$0.514531\pi$
$710$	0	0
$711$	−8928.00	−0.470923
$712$	0	0
$713$	−32550.0	−1.70969
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5832.00	0.303766
$718$	0	0
$719$	−18510.0	−0.960093	−0.480046	−	0.877243i	$-0.659380\pi$
$719$	−18510.0	−0.960093	−0.480046	+	0.877243i	$0.659380\pi$
$720$	0	0
$721$	18676.0	0.964675
$722$	0	0
$723$	2571.00	0.132250
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1009.00	0.0514742	0.0257371	−	0.999669i	$-0.491807\pi$
$727$	1009.00	0.0514742	0.0257371	+	0.999669i	$0.491807\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	33774.0	1.70886
$732$	0	0
$733$	−21994.0	−1.10828	−0.554138	−	0.832425i	$-0.686952\pi$
$733$	−21994.0	−1.10828	−0.554138	+	0.832425i	$0.686952\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24090.0	−1.20403
$738$	0	0
$739$	13948.0	0.694297	0.347148	−	0.937810i	$-0.387150\pi$
$739$	13948.0	0.694297	0.347148	+	0.937810i	$0.387150\pi$
$740$	0	0
$741$	−12963.0	−0.642655
$742$	0	0
$743$	26508.0	1.30886	0.654431	−	0.756122i	$-0.272908\pi$
$743$	26508.0	1.30886	0.654431	+	0.756122i	$0.272908\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7614.00	0.372934
$748$	0	0
$749$	−32568.0	−1.58880
$750$	0	0
$751$	1600.00	0.0777428	0.0388714	−	0.999244i	$-0.487624\pi$
$751$	1600.00	0.0777428	0.0388714	+	0.999244i	$0.487624\pi$
$752$	0	0
$753$	11772.0	0.569715
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30101.0	1.44523	0.722615	−	0.691250i	$-0.242940\pi$
$757$	30101.0	1.44523	0.722615	+	0.691250i	$0.242940\pi$
$758$	0	0
$759$	−13500.0	−0.645611
$760$	0	0
$761$	35628.0	1.69713	0.848564	−	0.529093i	$-0.177468\pi$
$761$	35628.0	1.69713	0.848564	+	0.529093i	$0.177468\pi$
$762$	0	0
$763$	1265.00	0.0600211
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7830.00	0.368611
$768$	0	0
$769$	−12517.0	−0.586963	−0.293482	−	0.955965i	$-0.594814\pi$
$769$	−12517.0	−0.586963	−0.293482	+	0.955965i	$0.594814\pi$
$770$	0	0
$771$	−8532.00	−0.398538
$772$	0	0
$773$	14124.0	0.657186	0.328593	−	0.944472i	$-0.393426\pi$
$773$	14124.0	0.657186	0.328593	+	0.944472i	$0.393426\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−10074.0	−0.465126
$778$	0	0
$779$	23244.0	1.06907
$780$	0	0
$781$	−19800.0	−0.907170
$782$	0	0
$783$	−6318.00	−0.288361
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−40433.0	−1.83136	−0.915680	−	0.401907i	$-0.868347\pi$
$787$	−40433.0	−1.83136	−0.915680	+	0.401907i	$0.868347\pi$
$788$	0	0
$789$	18180.0	0.820311
$790$	0	0
$791$	−32292.0	−1.45154
$792$	0	0
$793$	7975.00	0.357126
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27300.0	−1.21332	−0.606660	−	0.794962i	$-0.707491\pi$
$797$	−27300.0	−1.21332	−0.606660	+	0.794962i	$0.707491\pi$
$798$	0	0
$799$	−2340.00	−0.103609
$800$	0	0
$801$	−13392.0	−0.590740
$802$	0	0
$803$	−19380.0	−0.851688
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11718.0	0.511144
$808$	0	0
$809$	−2856.00	−0.124118	−0.0620591	−	0.998072i	$-0.519767\pi$
$809$	−2856.00	−0.124118	−0.0620591	+	0.998072i	$0.519767\pi$
$810$	0	0
$811$	12619.0	0.546379	0.273189	−	0.961960i	$-0.411921\pi$
$811$	12619.0	0.546379	0.273189	+	0.961960i	$0.411921\pi$
$812$	0	0
$813$	−6432.00	−0.277466
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−64517.0	−2.76275
$818$	0	0
$819$	−6003.00	−0.256119
$820$	0	0
$821$	−29082.0	−1.23626	−0.618130	−	0.786076i	$-0.712109\pi$
$821$	−29082.0	−1.23626	−0.618130	+	0.786076i	$0.712109\pi$
$822$	0	0
$823$	−10235.0	−0.433499	−0.216749	−	0.976227i	$-0.569545\pi$
$823$	−10235.0	−0.433499	−0.216749	+	0.976227i	$0.569545\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26976.0	−1.13428	−0.567139	−	0.823622i	$-0.691950\pi$
$827$	−26976.0	−1.13428	−0.567139	+	0.823622i	$0.691950\pi$
$828$	0	0
$829$	37802.0	1.58374	0.791868	−	0.610692i	$-0.209109\pi$
$829$	37802.0	1.58374	0.791868	+	0.610692i	$0.209109\pi$
$830$	0	0
$831$	6963.00	0.290666
$832$	0	0
$833$	14508.0	0.603448
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5859.00	0.241955
$838$	0	0
$839$	16974.0	0.698460	0.349230	−	0.937037i	$-0.386443\pi$
$839$	16974.0	0.698460	0.349230	+	0.937037i	$0.386443\pi$
$840$	0	0
$841$	30367.0	1.24511
$842$	0	0
$843$	−20466.0	−0.836164
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9913.00	0.402143
$848$	0	0
$849$	−12147.0	−0.491029
$850$	0	0
$851$	−21900.0	−0.882165
$852$	0	0
$853$	−24937.0	−1.00097	−0.500485	−	0.865745i	$-0.666845\pi$
$853$	−24937.0	−1.00097	−0.500485	+	0.865745i	$0.666845\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15756.0	0.628022	0.314011	−	0.949419i	$-0.398327\pi$
$857$	15756.0	0.628022	0.314011	+	0.949419i	$0.398327\pi$
$858$	0	0
$859$	−38144.0	−1.51508	−0.757542	−	0.652787i	$-0.773600\pi$
$859$	−38144.0	−1.51508	−0.757542	+	0.652787i	$0.773600\pi$
$860$	0	0
$861$	10764.0	0.426058
$862$	0	0
$863$	−5448.00	−0.214892	−0.107446	−	0.994211i	$-0.534267\pi$
$863$	−5448.00	−0.214892	−0.107446	+	0.994211i	$0.534267\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3513.00	0.137610
$868$	0	0
$869$	−29760.0	−1.16172
$870$	0	0
$871$	−23287.0	−0.905913
$872$	0	0
$873$	−2871.00	−0.111304
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21191.0	0.815928	0.407964	−	0.912998i	$-0.366239\pi$
$877$	21191.0	0.815928	0.407964	+	0.912998i	$0.366239\pi$
$878$	0	0
$879$	6714.00	0.257631
$880$	0	0
$881$	18216.0	0.696609	0.348305	−	0.937381i	$-0.386758\pi$
$881$	18216.0	0.696609	0.348305	+	0.937381i	$0.386758\pi$
$882$	0	0
$883$	−12767.0	−0.486573	−0.243286	−	0.969955i	$-0.578225\pi$
$883$	−12767.0	−0.486573	−0.243286	+	0.969955i	$0.578225\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11010.0	−0.416775	−0.208388	−	0.978046i	$-0.566822\pi$
$887$	−11010.0	−0.416775	−0.208388	+	0.978046i	$0.566822\pi$
$888$	0	0
$889$	29440.0	1.11067
$890$	0	0
$891$	2430.00	0.0913671
$892$	0	0
$893$	4470.00	0.167506
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−13050.0	−0.485760
$898$	0	0
$899$	−50778.0	−1.88381
$900$	0	0
$901$	−43056.0	−1.59201
$902$	0	0
$903$	−29877.0	−1.10105
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−22772.0	−0.833662	−0.416831	−	0.908984i	$-0.636859\pi$
$907$	−22772.0	−0.833662	−0.416831	+	0.908984i	$0.636859\pi$
$908$	0	0
$909$	−7128.00	−0.260089
$910$	0	0
$911$	−29802.0	−1.08385	−0.541923	−	0.840428i	$-0.682304\pi$
$911$	−29802.0	−1.08385	−0.541923	+	0.840428i	$0.682304\pi$
$912$	0	0
$913$	25380.0	0.919995
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11040.0	0.397571
$918$	0	0
$919$	−48941.0	−1.75671	−0.878354	−	0.478011i	$-0.841358\pi$
$919$	−48941.0	−1.75671	−0.878354	+	0.478011i	$0.841358\pi$
$920$	0	0
$921$	−4155.00	−0.148656
$922$	0	0
$923$	−19140.0	−0.682558
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−7308.00	−0.258928
$928$	0	0
$929$	31026.0	1.09573	0.547863	−	0.836568i	$-0.315441\pi$
$929$	31026.0	1.09573	0.547863	+	0.836568i	$0.315441\pi$
$930$	0	0
$931$	−27714.0	−0.975607
$932$	0	0
$933$	17010.0	0.596873
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11183.0	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$937$	11183.0	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$938$	0	0
$939$	−1263.00	−0.0438940
$940$	0	0
$941$	−2562.00	−0.0887554	−0.0443777	−	0.999015i	$-0.514130\pi$
$941$	−2562.00	−0.0887554	−0.0443777	+	0.999015i	$0.514130\pi$
$942$	0	0
$943$	23400.0	0.808069
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7638.00	−0.262093	−0.131046	−	0.991376i	$-0.541834\pi$
$947$	−7638.00	−0.262093	−0.131046	+	0.991376i	$0.541834\pi$
$948$	0	0
$949$	−18734.0	−0.640813
$950$	0	0
$951$	29952.0	1.02130
$952$	0	0
$953$	51432.0	1.74821	0.874106	−	0.485735i	$-0.161448\pi$
$953$	51432.0	1.74821	0.874106	+	0.485735i	$0.161448\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−21060.0	−0.711362
$958$	0	0
$959$	6486.00	0.218398
$960$	0	0
$961$	17298.0	0.580645
$962$	0	0
$963$	12744.0	0.426448
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−39728.0	−1.32116	−0.660582	−	0.750754i	$-0.729691\pi$
$967$	−39728.0	−1.32116	−0.660582	+	0.750754i	$0.729691\pi$
$968$	0	0
$969$	−34866.0	−1.15589
$970$	0	0
$971$	47946.0	1.58461	0.792307	−	0.610123i	$-0.208880\pi$
$971$	47946.0	1.58461	0.792307	+	0.610123i	$0.208880\pi$
$972$	0	0
$973$	36892.0	1.21552
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22326.0	−0.731087	−0.365544	−	0.930794i	$-0.619117\pi$
$977$	−22326.0	−0.731087	−0.365544	+	0.930794i	$0.619117\pi$
$978$	0	0
$979$	−44640.0	−1.45730
$980$	0	0
$981$	−495.000	−0.0161102
$982$	0	0
$983$	48468.0	1.57262	0.786312	−	0.617830i	$-0.211988\pi$
$983$	48468.0	1.57262	0.786312	+	0.617830i	$0.211988\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2070.00	0.0667567
$988$	0	0
$989$	−64950.0	−2.08826
$990$	0	0
$991$	25141.0	0.805883	0.402942	−	0.915226i	$-0.367988\pi$
$991$	25141.0	0.805883	0.402942	+	0.915226i	$0.367988\pi$
$992$	0	0
$993$	12684.0	0.405352
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−35422.0	−1.12520	−0.562601	−	0.826729i	$-0.690199\pi$
$997$	−35422.0	−1.12520	−0.562601	+	0.826729i	$0.690199\pi$
$998$	0	0
$999$	3942.00	0.124844

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.v.1.1		1
4.3	odd	2		150.4.a.g.1.1	yes	1
5.2	odd	4		1200.4.f.q.49.1		2
5.3	odd	4		1200.4.f.q.49.2		2
5.4	even	2		1200.4.a.r.1.1		1
12.11	even	2		450.4.a.i.1.1		1
20.3	even	4		150.4.c.b.49.1		2
20.7	even	4		150.4.c.b.49.2		2
20.19	odd	2		150.4.a.c.1.1	✓	1
60.23	odd	4		450.4.c.h.199.2		2
60.47	odd	4		450.4.c.h.199.1		2
60.59	even	2		450.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.a.c.1.1	✓	1	20.19	odd	2
150.4.a.g.1.1	yes	1	4.3	odd	2
150.4.c.b.49.1		2	20.3	even	4
150.4.c.b.49.2		2	20.7	even	4
450.4.a.i.1.1		1	12.11	even	2
450.4.a.l.1.1		1	60.59	even	2
450.4.c.h.199.1		2	60.47	odd	4
450.4.c.h.199.2		2	60.23	odd	4
1200.4.a.r.1.1		1	5.4	even	2
1200.4.a.v.1.1		1	1.1	even	1	trivial
1200.4.f.q.49.1		2	5.2	odd	4
1200.4.f.q.49.2		2	5.3	odd	4

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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1200.4.a.v.1.1

Level	$1200$
Weight	$4$
Character	1200.1
Self dual	yes
Analytic conductor	$70.802$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$